Question

Suppose the data have a bell-shaped distribution with a mean of 30 and a standard deviation of 5. Use the empirical rule to determine the percentage of data within each of the following ranges:

a. 20 to 40
b. 15 to 45
c. 25 to 35

Answer 1

a) 95% of the data fall in the range of 20 to 40.

b) 99.7% of the data fall in the range of 15 to 45.

c) 68% of the data fall in the range of 25 to 35.

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 30

Standard deviation = 5

a. 20 to 40

20 = 30 - 2*5

So 20 is two standard deviations below the mean

40 = 30 + 2*5

So 40 is two standard deviations above the mean

By the Empircal rule, 95% of the data fall in the range of 20 to 40.

b. 15 to 45

15 = 30 - 3*5

So 15 is three standard deviations below the mean

45 = 30 + 3*5

So 45 is three standard deviations above the mean

By the Empircal rule, 99.7% of the data fall in the range of 15 to 45.

c. 25 to 35

25 = 30 - 1*5

So 25 is one standard deviation below the mean

35 = 30 + 1*5

So 35 is three standard deviation above the mean

By the Empirical rule, 68% of the data fall in the range of 25 to 35.